| L(s) = 1 | + 9·9-s + 18·11-s − 14·19-s + 10·25-s + 12·29-s + 22·31-s − 18·41-s + 49-s − 24·59-s − 2·61-s − 6·71-s − 4·79-s + 44·81-s + 12·89-s + 162·99-s − 24·101-s + 10·109-s + 161·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3·9-s + 5.42·11-s − 3.21·19-s + 2·25-s + 2.22·29-s + 3.95·31-s − 2.81·41-s + 1/7·49-s − 3.12·59-s − 0.256·61-s − 0.712·71-s − 0.450·79-s + 44/9·81-s + 1.27·89-s + 16.2·99-s − 2.38·101-s + 0.957·109-s + 14.6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.20287718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.20287718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | | \( 1 \) | |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 37 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) | 4.3.a_aj_a_bl |
| 7 | $D_4\times C_2$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_ab_a_ad |
| 11 | $C_4$ | \( ( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.as_gh_abka_fmn |
| 13 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 833 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_abt_a_bgb |
| 17 | $D_4\times C_2$ | \( 1 - 33 T^{2} + 569 T^{4} - 33 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_abh_a_vx |
| 19 | $C_4$ | \( ( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.o_ex_bfg_gbx |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.am_dc_axo_ghq |
| 31 | $C_4$ | \( ( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.aw_kx_adqu_xqv |
| 37 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 1518 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abo_a_cgk |
| 41 | $D_{4}$ | \( ( 1 + 9 T + 101 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.s_kx_dui_bdxd |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) | 4.43.a_a_a_dby |
| 47 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 494 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abw_a_ta |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ahw_a_xsg |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.59.y_rk_hoq_csnm |
| 61 | $D_{4}$ | \( ( 1 + T + 111 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.c_ip_ng_bdkv |
| 67 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 23198 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ajg_a_biig |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 143 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.g_lj_bxk_bvbl |
| 73 | $D_4\times C_2$ | \( 1 + 36 T^{2} - 538 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_bk_a_aus |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.e_iy_bds_bmqg |
| 83 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 27998 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ajg_a_bpkw |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.am_mi_aecq_ckti |
| 97 | $D_4\times C_2$ | \( 1 + 39 T^{2} + 17297 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_bn_a_zph |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.66581620143539781578983484820, −6.29720852306682266490098189385, −6.29179340046815438745998539320, −6.28413194147013180876171839497, −6.19887520711151453972069857347, −5.51036736273608631414599207964, −5.21023822604839441202975886528, −4.64626836929443240811067997655, −4.62806068679460302769009863625, −4.58689221081780799182260439810, −4.34425528069808022455527823395, −4.32487028003837155697461268750, −4.16650434119763488993331408300, −3.78083280748645323801804237318, −3.63151586006120526617734760802, −3.15955995242963240899943542471, −3.04674386223366010725449921372, −2.85545714268729118782634492363, −2.24311497210516514057745001151, −1.72797408067733927151121193433, −1.67960189489757914005721342674, −1.64866083417210551987370985058, −1.16496266223136356532337259221, −0.894142865497532192362774522542, −0.77518004565332744831613777496,
0.77518004565332744831613777496, 0.894142865497532192362774522542, 1.16496266223136356532337259221, 1.64866083417210551987370985058, 1.67960189489757914005721342674, 1.72797408067733927151121193433, 2.24311497210516514057745001151, 2.85545714268729118782634492363, 3.04674386223366010725449921372, 3.15955995242963240899943542471, 3.63151586006120526617734760802, 3.78083280748645323801804237318, 4.16650434119763488993331408300, 4.32487028003837155697461268750, 4.34425528069808022455527823395, 4.58689221081780799182260439810, 4.62806068679460302769009863625, 4.64626836929443240811067997655, 5.21023822604839441202975886528, 5.51036736273608631414599207964, 6.19887520711151453972069857347, 6.28413194147013180876171839497, 6.29179340046815438745998539320, 6.29720852306682266490098189385, 6.66581620143539781578983484820
